The diameter of the Moon is approximately one-fourth of the diameter of the Earth. What is the ratio (approximate) of their volumes?

Options

A. `4:16`
B. `1:64`
C. `VA`
D. `1:128`

Answer

To find the ratio of the volumes of the Moon and the Earth, we can follow these steps:

### Step 1: Understand the relationship between diameters and radii
The diameter of the Moon is approximately one-fourth of the diameter of the Earth. If we denote the diameter of the Earth as \( D_E \) and the diameter of the Moon as \( D_M \), we can express this relationship as:
\[ D_M = \frac{1}{4} D_E \]

### Step 2: Find the radii of the Earth and the Moon
The radius is half of the diameter. Therefore, the radius of the Earth \( R_E \) and the radius of the Moon \( R_M \) can be expressed as:
\[ R_E = \frac{D_E}{2} \]
\[ R_M = \frac{D_M}{2} = \frac{1}{2} \left( \frac{1}{4} D_E \right) = \frac{1}{8} D_E \]

### Step 3: Write the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 4: Calculate the volumes of the Earth and the Moon
Using the formula for volume, we can express the volumes of the Earth and the Moon as:
\[ V_E = \frac{4}{3} \pi R_E^3 = \frac{4}{3} \pi \left( \frac{D_E}{2} \right)^3 \]
\[ V_M = \frac{4}{3} \pi R_M^3 = \frac{4}{3} \pi \left( \frac{D_E}{8} \right)^3 \]

### Step 5: Simplify the expressions for the volumes
Now, let's simplify the volumes:
- For the Earth:
\[ V_E = \frac{4}{3} \pi \left( \frac{D_E}{2} \right)^3 = \frac{4}{3} \pi \left( \frac{D_E^3}{8} \right) = \frac{4 \pi D_E^3}{24} = \frac{\pi D_E^3}{6} \]

- For the Moon:
\[ V_M = \frac{4}{3} \pi \left( \frac{D_E}{8} \right)^3 = \frac{4}{3} \pi \left( \frac{D_E^3}{512} \right) = \frac{4 \pi D_E^3}{1536} = \frac{\pi D_E^3}{384} \]

### Step 6: Find the ratio of the volumes
Now we can find the ratio of the volumes of the Moon to the Earth:
\[ \text{Ratio} = \frac{V_M}{V_E} = \frac{\frac{\pi D_E^3}{384}}{\frac{\pi D_E^3}{6}} \]

### Step 7: Simplify the ratio
The \( \pi D_E^3 \) terms cancel out:
\[ \text{Ratio} = \frac{1}{384} \times \frac{6}{1} = \frac{6}{384} = \frac{1}{64} \]

### Conclusion
Thus, the approximate ratio of the volumes of the Moon to the Earth is:
\[ \text{Ratio} = 1 : 64 \]

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